This might sound like a strange question, but here goes...
When men urinate, there is often some "nitzotzos" that get on the toilet seat or elsewhere and need to be cleaned. On Shabbos, this can be difficult to do quickly and well (the latter being absolutely necesary in some circumstances!).
Can you discuss the issues involved and the best way to approach it?
Answer:
The question is not strange, and perfectly legitimate.
In practice, the "nitzvotzos" do not involve a great volume of fluid, and therefore there is no problem in taking a piece or two of toilet paper, and wiping the toilet seat clean, as one would do in the weekdays. There is no reason why this can't be done just as well on Shabbos as on weekdays.
In general, it is good training and good chinnuch for manners and consideration to raise the toilet seat before use. This is true even without the Shabbos issue you raise!